Equations of Lines TN
Aims and Objectives
The teacher can decide to take longer over page one and two of the worksheets, where students come up with their own definitions for the difference between any two lines, or they may choose to limit the time on them or cut them completely because they want the class to practise gradient and y-intercepts. The aim of this resource is to provide the resources to support a number of different approaches and allow a great deal of flexibility into what task/activity different students of differing abilities are asked to work on at any one time, the applets offering feedback and guidance. The focus is that ALL students are able, by the end of one or two hours, to find the equation of a line from its gradient and y-intercept, on paper.
Towards the end of the lesson it can be helpful to use Geogebra's "record to spreadsheet" tool to show how the gradient and y-intercept perfectly defines the x,y coordinates for all points on a line. Many students miss this connection. It's also really helpful when many students opt for the x-intercept as a means of differentiating between parallel lines. Whilst it's true each parallel line has a unique x-intercept, it would be difficult (impossible?) to use this definition to derive an equation that maps the x-coordinate of every point on the line, to its corresponding y-coordinate.
It's these sort of rich discussions that help students to understand that mathematics is a thoughtful response to problems humans have needed to solve - not an unnecessarily complicated solution to simple problems, designed by one group of people to confuse and mislead another!
Student Solutions to the Problem of Defining a Line
Below are some examples of solutions students have come it with on their own, without any prior knowledge of gradients and y-intercepts. These options provide a great opportunity for questioning: "what about this line? (the teacher draws a line that will challenge the students model)", reflection, class debate and consensus, to see if we can refine their models to greater and greater degrees of precision. It also allows the class to develop some ground rules as to the key differences between lines and what an effective system of classification might look like.
Experience from the classroom
I’ve found that students who have little/no confidence in maths often need to know that they are “getting it right” to reassure them and make them feel comfortable with a task, or teacher. When they experience success with a given teacher then they’ll believe they can be successful in other tasks that that teacher may give them. Subsequently the teacher can introduce greater degrees of investigation and speculation in the activities they offer that student.
I’ve also found some students, however able they may be, have decided they are only going to give their mathematical studies “x” amount of time. If I ask them to investigate and explore, we sometimes do not have time to draw the different threads of their ideas together into a coherent whole, or we don’t get time both for them to create their own mathematics and to gradually introduce them to the internationally recognised definitions and representations of the mathematics being investigated. I want them to think mathematically and get involved, they want to get the best grade possible in the minimum amount of time. They may feel that the investigative approach being offered isn’t the most effective one for achieving their goals. We may not get time to cover as much of the syllabus as they think we should have done or as quickly as they would have liked, so as to leave plenty of time at the end for them to practice and integrate the mathematics that we’ve covered. They start to get anxious, frustrated, or worse, lose confidence in their ability to succeed with the teaching approach used.
Some students find technology really helpful in visualising and offering feedback on their mathematics. Other students find technology confusing and an additional “thing to have to learn to use” i.e. the technical side of using the computer distracts (and can intimidate) them. They may emotionally detach themselves from, or rebel against, the mathematics they are trying to learn.
This resource is designed to try and meet the needs of all these types of students and the varying range of goals students, and their families, might fix for themselves.
It would be great to get your thoughts and feedback: firstname.lastname@example.org.